The Progressives were reformers in the late 19th and early 20th century who believed that in order to address modern problems, America needed to abandon the old ideas of the Founding in favor of a new expansive conception of the role of government. Progressives paved the way for modern liberalism and politics, and their core ideas are still the mainstay of today’s liberalism.

Actually the progressives were building a new more expansive FEDERAL government to oppose the corruption and power of PRIVATE governments run by monopolies and corrupt city and State governments dominated by wealth and private interest. It was said in the 1800's that the Penn Railroad ran Pennsylvania and Vanderbilt controlled New York State. So this framing is designed to sell a product not tell the truth. It is basically an effort of the rich and powerful to rewrite the historical narrative.

They then go on to list all the heroes of the period;

Some Progressives were prominent journalists such as Herbert Croly (co-founder of The New Republic), some were distinguished professors such as John Dewey and Woodrow Wilson (president of Princeton before he was President of the U.S.), and many were political leaders such as Theodore Roosevelt and Robert La Follette. Progressives could be found in both political parties: Wilson was a Democrat, Roosevelt was a Republican.

And they both try to hook people with misrepresentations of what the progressives were about, and at the same time manage to defame all the heroes of the progressive era (including people they otherwise claim to admire) with a broad brush:

The Progressives were united in their contempt for what they called the “individualism” of the Founding. Instead of a government that protects natural rights through limited, decentralized powers, they envisioned an expansive government, a “living” and evolving Constitution, and the rule of “experts” in nationally centralized administrative agencies.

Contempt for individualism

Actually their contempt was for the abusiveness of the "Horatio Alger" narrative of the lucky few pulling themselves up by non-existent bootstraps; when the reality is that most people simply wanted to make a living, improve the lives of their children and leave the world a better place for their grandchildren. The Horatio Alger myth creates a world of a one percent of "winners" and brands the other 99% as losers. And since the "economic royalists" [FDR's term] of the previous century were utterly corrupt, willing to recklessly go to war with one another [When Rockefeller laid his pipeline network he brought down the entire economy and many of the weaker railroads just to undermine his erstwhile allies in the rail industry], and owning the Senate and Courts, theirs wasn't a struggle against "individualism" but against corporate Kinglets.

Natural Rights

And DeMint turns the "natural rights" narrative on it's head too. The progressives were not against natural rights, on the contrary they sought to use the Federal Government to make sure that individuals could retain their natural rights in the face of corporate kleptocracy, rent seeking, legal theft of mineral rights and land, and monopoly. To do this they had to use "Hamiltonian methods" to advance "Jeffersonian Goals." TR would first use the power of anti-trust, and when that wasn't sufficient he came up with the idea of organizing corporations so that they could be better governed. TR's ideas never really were fully realized, and Reagan corruptly ended efforts at Anti-Trust, so the issue now remains the same for modern day Progressives. The Empire Struck back, through it's armies of attorneys. And by creating the Heritage Foundation and then corrupting it into a propaganda rag. They've not only created an astroturfed ideology, but an alternative reality. A kind of corporate version of "Game of Thrones" only with lawyers instead of armed warriors.

Living Constitution

Most of the right would also have you believe that all their precepts are based on the founding fathers, who somehow in their narrative were all rugged individualists who interpreted the constitution exactly the same and collectively wanted it frozen in ice (except where most conservatives want to chip away at it). They claim to want a literal and strict reading of the constitution, but since they ignore (conveniently) words they don't like they ignore the "militia" clause of the second amendment and ignore the 18th century meaning of "secure in one's person" since the word "privacy" in those days referred to Privy's (urinals) not privacy as we understand it now and "secure in one's person" referred to privacy. Since they are only literal or "strict" where it suits their convenience what they really want is an authoritarian or even tyrannical reading of the constitution that affirms their private separate advantage to make judgments and rule people locally with impunity. This puts the lie to their argument that it was "progressives" who wanted an expansive government. The conservatives just want that government either bought and paid for by themselves, or delegated to corporate and landlord rule. In short the argument isn't between a living constitution (since the advocates also interpret it) or a strictly interpreted one, but between a living constitution and the dead hand of local and distributed tyranny.

Experts

The one accusation that I'll grant the DeMint has a point on. And that Progressives made, was that they settled on administrative (bureaucratic) efforts to rule American Business. They started with experts, but nowadays they are likely to be well connected lawyers and the experts are either marginalized or get fired if they open their mouths. Still I'd rather have a genuinely neutral expert playing judge and having to justify his decisions than pretend creationists who are really Social Darwinists... The progressives did have faults. Just not the ones outlined by DeMint's heritage foundation.

And unfortunately regulation has become so complicated and such a round robin "captured" endeavor, that the government lawyers writing the rules are often working almost directly for the regulated, and they use their expertise to make money for the corporations once they leave government. Conservatives only pretend to complain about rent seeking, since that is the real object of de-regulation. If the Government doesn't regulate business, they locate businesses where they can kill the most [poor] people when something goes wrong -- like West Texas and then want the taxpayer to pick up the Tab, and "Gubbornment" to take the blame -- as if they weren't in fact doing much of the governing.

This adamantine romanticization of the 19th century (apparently including Antebellum Southern Slavery) is the real problem with the right in general and the Heritage Foundation and it's other organs. Folks like Jason Richwine, can't help themselves in exposing their racism, as he did in his College Thesis, because their sense of entitlement is on their sleeves and they regularly isolate themselves from reality. Prejudice is judging things before one actually studies all the angles, and that is what these articles represent. When one believes something "in advance" of evidence, one hunts for any argument to advance one's "truth".